Henderson has a beach, nothing really to dive though, just sand. The campsites are all the RV type spots, think peagravel bottomed campsites.
Big Lagoon is adjascent to the intercoastal waterway, not really anything to dive there either, but its close to the bridge, which can apparently make a pretty decent dive at times. Short drive out of the park.
Across the intercoastal from Big Lagoon is Johnson Beach, it has primitive camping only and requires a hike to get to where you are allowed to camp. There's diving on the very end of the the beach, near Fort McRae, in the form of steel jetties. I promise you that you won't want to walk your gear that far(several miles of sand).
St. Andrews and Ft Pickens are the only campsites I know of with beach dives attached to them in this area.
Dunno the rules at the new Alabama State Pier, but if they allow beach diving, you could always camp across the street at the State Park in Gulf Shores?